Surface Plasmon Polaritons

Justin White
March 19, 2007

(Submitted as coursework for AP272, Stanford University,
Winter 2007)

Surface plasmon polaritons are collective
longitudinal oscillations of electrons near a material surface, strongly
coupled to an electromagnetic wave. The existence of coherent electron
oscillations bound to the surface of a conductor were first predicted by
Ritchie[1] in 1957 and demonstrated by Powell and Swan[2,3] in 1959, but
have recently experienced an explosion of interest due to their ability
to efficiently manipulate light on a deep sub-wavelength scale. Because
the electromagnetic wave is coupled to the motion of conduction band
electrons bound to the material surface, the fields are strongly
localized to the surface, opening up possibilities of sub-wavelength
optical detectors and waveguides; compact, sensitive chemical detectors;
and enhanced light-matter interaction.

Bulk and Surface Plasmons

Both bulk and surface plasmons have associated
electromagnetic waves, and can consequently be described by Maxwell's
equations. The coherent oscillations of electron motion can be
encapsulated in the dielectric constant of the material. The derivation
of the electromagnetic fields which characterize bulk and surface
plasmons is done below, and those interested are encouraged to work out
the results. To keep things brief however, the basic form of the bulk
and surface plasmon solutions are shown below, along with a plot of the
dispersion relation for these modes.

The dispersion diagram relates the time-variation of
the wave (given by its frequency &omega) to the spatial variation of the
wave (given by its wave-vector kx). Bulk plasmons are
associated with purely transverse electromagnetic waves, with the
electric and magnetic fields perpendicular to the direction of
propagation, and can only exist for &omega > &omegap (the
plasma frequency). &omegap is the resonant frequency of free
electrons in response to a perturbation (such as an electromagnetic
field). For &omega < &omegap, the wave-vector for bulk
plasmons becomes imaginary, giving an exponentially decaying wave
instead of a propagating wave. It is for this reason that most metals
are highly reflective for visible light (&omega < &omegap
&asymp 10eV), but transparent for ultraviolet light (&omega >
&omegap) [4, pg. 275].

Surface plasmons have an associated electromagnetic
wave with both transverse and longitudinal field components. Such waves
can only be excited at the interface between a conductor and dielectric,
and are tightly bound to the surface. The fields reach their maximum at
the interface (z=0), and exponentially decay away from the surface. The
wave-vector of the surface plasmon mode (kspp) always
lies to the right of the free space wave-vector (ko),
such that &lambdaspp < &lambdao, where
&lambdaspp is the wavelength of the surface plasmon and
&lambdao is the wavelength of light in free space (vacuum).
Additionally, this makes it impossible to directly launch a surface
plasmon wave by illumination with free-space radiation: the free-space
photons simply do not have enough momentum to excite the surface
plasmon. As &omega increases, kspp gets larger and
larger, moving further away from ko. As
kspp increases, the surface plasmon wavelngth
decreases and the wave is more tightly bound to the surface. This
process has an upper limit of &omegasp, the surface plasmon
resonant frequency, which occurs when the dielectric constant of the
metal and the dielectric have the same magnitude but opposite signs.

Excitation of Surface Plasmons

Surface plasmons were first predicted by Ritchie[1]
in 1957, and were experimentally verified by Powell and Swan[2,3] in
1959 with electron energy loss spectroscopy measurements. In these
measurements, high energy electrons were used to bombard a thin metallic
film and launch surface plasmons. The charge of the high energy
electrons couples to the plasma oscillations of electrons in the metal;
by tuning the energy and angle of incidence of the electrons, the
wave-vector can be tuned and surface plasmons of a whole range of
wavelengths can be excited. However, due to the high energy of the
electrons, they have a very large momentum and it is difficult to
precisely control the coupling: even a slight spread in energy or angle
will result in a very broad range of surface plasmon wavelengths being
excited. Additionally, only surface plasmons far along the dispersion
curve, where kspp is largest, are generally excited.
A more precise method to launch surface plasmons is through coupling
with an incident electromagnetic wave (photon).

As mentioned previously, direct excitation of surface
plasmons by free-space photons is not achievable because
kspp is always greater than ko; this
can be seen in figure 1, where the surface plasmon dispersion relation
always lies to the right of the free space dispersion curve.

This can be overcome by back-side illumination
through a material with a higher index of refraction (n), where
the far-field radiation has a larger wave-vector
(k=nko). This type of coupling geometry is
shown in figure 2, and is known as a Kretschmann-Raether coupler[5]. A
surface plasmon will be efficiently excited when k|| =
nkosin(&theta) = kspp. Kretschmann
couplers are commonly used in experiments, but are limited to very thin
films such that the high-k photon is able to tunnel through the film and
couple to a surface plasmon on the lower-index surface.

A more general approach to launching surface plasmons
with light is the use of structured surfaces that are able to impart
momentum on the photon, enabling it to couple to the surface plasmon
mode. Anything from a single sub-wavelength disk or slit, to
rectangular or sinusoidal diffraction gratings are used for this type of
coupling. A thorough overview of surface plasmon coupling and patterned
and rough surfaces is given by Raether[6].

Applications of Surface Plasmons

Surface plasmons have many interesting applications
such as optical measurements of films, chemicals, and biological agents
bound to a metallic surface; confinement and guiding of light below the
classical diffraction limit; and enhanced light-matter interaction such
as surface-enhanced Raman spectroscopy (SERS). A few of the more
interesting applications from the literature and industry are outlined
below.

Surface plasmon resonance sensors based on
Kretschmann couplers can be used to make precise optical measurements at
a metal-dielectric interface. The basic idea behind these detectors is
encapsulated in the coupling condition discussed above for the
Kretschmann geometry: k|| =
nkosin(&theta) = kspp. Since the
surface plasmon is tightly bound to the metal-dielectric interface, any
changes in that interface will translate directly into changes in the
surface plasmon wave-vector kspp. This will cause a
corresponding change in the optimal excitation angle for the Kretschmann
coupler; by measuring this angle, the change in kspp
can be deduced, allowing for the change in the refractive index to be
computed. These detectors are sensitive enough that they can measure a
change in film thickness of just a single monolayer, or refraction index
changes (for example, due to chemical binding events on a
bio-functionalized gold surface) as small as 10-6 RIU
(refractive index units). For example, the commercial sensiQ detector
from Nomadics is capable of measuring binding events of just one
picogram of protein/mm2[7].

More recently, there has been an explosion of
interest in surface plasmons on nanoscale patterned surfaces. Using
modern nanofabrication facilities, it is possible to make structures
with feature sizes much smaller than the wavelength of visible light
(&sim 500nm). For example, enhanced transmission through arrays of
sub-wavelength slits has been experimentally demonstrated[8] and has
promising applications in photolithography and high-density optical data
storage. Because surface plasmons can be launched at optical
frequencies with much smaller wavelengths, it is possible to manipulate
the flow of light on the scale of tens of nanometers. This has led to
the coining of the term "plasmonics"[9], where surface plasmons are used
to manipulate and transport information, analogous to electronics and
photonics. Photonics has nearly unlimited bandwidth and almost no loss
or dispersion, making it invaluable for high-throughput long-haul
information networks. However, photonics components are
diffraction-limited to micron-scale sizes, ruling out dense integration
with nanoscale electronics. Plasmonics is potentially poised to bridge
the tens of nanometer scale of modern electronics to the micron scale
world of photonics. Many interesting devices, such as a 50nm
metal-insulator-metal (MIM) surface plasmon waveguide capable of guiding
1.5um light[10] even around 90o corners, have been proposed
and are being actively investigated. Such devices have great promise,
but also must overcome the limits imposed by the lossy propagation of
surface plasmons in metals. The propagation lengths of surface plasmon
waves in the visible and near infrared regimes are typically on the
order of hundreds of microns on unpatterned surfaces, but drop to under
thirty microns in sub-wavelength MIM waveguides. The propagation length
in a modern optical fiber is drastically larger, on the order of
kilometers. The small propagation lengths of surface plasmons are a
significant hurdle in the practical implementation of plasmonics;
however, researchers throughout the world are actively pursuing
geometries which take account for this loss and new designs to limit the
loss as much as possible.

Surface plasmons have many exciting potential
applications and rich underlying physics drawing heavily from
solid-state physics and electrodynamics. Although known for over 50
years, there are still many unsolved theoretical and experimental
problems in the field of surface plasmons. Additionally, with the use
of modern nanofabrication techniques, a whole range of new plasmonic
devices have become possible. Although not without their drawbacks,
surface plasmon based devices have a plethora of laboratory and
commercial applications--including the potential to revolutionize
photonic-electronic integration with the burgeouning field of
plasmonics.

Appendix: Derivation of Bulk and Surface Plasmons

Both bulk and surface plasmons have associated
electromagnetic waves, and can consequently be described by Maxwell's
equations. The coherent oscillations of electron motion can be
encapsulated in the dielectric constant of the material. A simple model
for the motion of an electron in a conductor is a free electron driven
by the field of the electromagnetic wave.

This electron motion can then be converted into a
dielectric constant by using the definitions of the dipole moment,
electric susceptibility, and dielectric constant[11]:

&omegap is known as the plasma frequency.
This dielectric constant can then be inserted into the standard
electromagnetic wave equation

The wave equation allows for the standard propagating
wave solution, given by

This is a valid solution provided that

This is the dispersion relation of bulk plasmon
polaritons propagating in a metallic film. The dispersion relation
gives the correspondence between the time-dependence of the
electromagnetic wave (&omega), and the spatial variation (k);

the wavelength of the wave is given by
&lambda=2&pi/k. Figure A-1 shows a plot of the bulk plasmon dispersion
relation (solid line), along with the free space dispersion relation
(&omega = ck). For frequencies above the plasma frequency
&omegap, the metal supports propagating modes. For
frequencies below &omegap, k becomes imaginary and the fields
exponentially decay inside the metal. For typical metals,
h&omegap/2&pi &asymp 10eV (in the ultra-violet); for
semiconductors, h&omegap/2&pi < 0.5eV (in the terahertz
regime). Although bulk plasmons, which can exist even deep inside a
metal, do not exist for &omega < &omegap, there is an
additional solution to Maxwell's equations for these frequencies:
surface plasmons.

Surface plasmons are propagating waves bound to
plasmon oscillations of electrons at a metal-dielectric interface. The
field intensity of the wave is at a maximum at the interface and
exponentially decays away from the interface. The mathematical form of
surface plasmons can be obtained by solving Maxwell's equations at a
metal-dielectric interface:

Assuming the metal-dielectric interface lies in the
xy plane at z = 0, it can be shown that x-propagating surface waves of
the following form are solutions to Maxwell's equations:

with Ex,m = Ex,d
(E|| continuous), &epsilonmEz,m =
&epsilondEz,d (D&perp continuous), and
Hy,m = Hy,d (B|| continuous); all other
field components are zero. The subscript j is used to denote
fields in the metal or dielectric. The dispersion relation for surface
plasmons can be obtained by inserting the equations for E and H into
Maxwell's equations and enforcing the boundary conditions:

The dispersion relation, assuming
&epsilonm = 1 - (&omegap/&omega)2, is
shown in Fig. 1, along with the free-space and bulk plasmon
dispersion relations. The surface plasmon dispersion relation always
lies to the right of the free-space dispersion relations; as a result,
free-space radiation does not have enough momentum to launch surface
plasmons.

Additionally, the magnitude of kx
for the surface plasmon asymptotically approaches &infin as &omega
approaches &omegasp, where &epsilonm =
-&epsilond. For &epsilonm = 1 -
(&omegap/&omega)2, &omegasp =
&omegap / &radic(1 + &epsilond). As
kx increases, the surface plasmon wavelength decreases
and &kappaz increases, resulting in a rapidly varying
surface wave that is more tightly confined to the surface. For smaller
values of &omega, the surface plasmon wave vector approaches the light
line, the surface plasmon and free-space wavelengths become more
similar, and the surface wave becomes less and less strongly confined to
the surface.

For the simple free-electron model considered thus
far, there is no loss in the system and the surface wave will have an
infinite propagation length. In real materials, however, there is loss
and the wave will only propagate a finite length. The propagation
length will decrease as the surface wave becomes more tightly bound to
the interface (as &omega approaches &omegasp), as the energy
of the field becomes more concentrated in the metal. Consequently,
there is a trade-off between confinement and propagation distance.

A final property of interest that can be obtained
from the dispersion diagram shown in figure A-2 is the group velocity of
the surface wave. Any information carried by the wave will propagate at
the wave group velocity, vg = (&part &omega / &part k). As
&omega approaches &omegasp, vg asymptotically
approaches zero.

The dispersion diagram plots shown above were
generated using
matplotlib, an open source
plotting library written in Python;
the source code of the python script for the plots can be downloaded here. The images of the above equations were
generated using latex and latex2html; the .tex file can be downloaded here. The figure of the Kretschmann-Raether
coupler was drawn in Inkscape, an
open source vector graphics editor; the svg file can be downloaded here.